299 research outputs found

### Distributional Borel Summability for Vacuum Polarization by an External Electric Field

It is proved that the divergent perturbation expansion for the vacuum
polarization by an external constant electric field in the pair production
sector is Borel summable in the distributional sense.Comment: 14 page

### Distributional Borel Summability of Odd Anharmonic Oscillators

It is proved that the divergent Rayleigh-Schrodinger perturbation expansions
for the eigenvalues of any odd anharmonic oscillator are Borel summable in the
distributional sense to the resonances naturally associated with the system

### Perturbation theory of PT-symmetric Hamiltonians

In the framework of perturbation theory the reality of the perturbed
eigenvalues of a class of \PTsymmetric Hamiltonians is proved using stability
techniques. We apply this method to \PTsymmetric unperturbed Hamiltonians
perturbed by \PTsymmetric additional interactions

### Canonical Expansion of PT-Symmetric Operators and Perturbation Theory

Let $H$ be any \PT symmetric Schr\"odinger operator of the type $-\hbar^2\Delta+(x_1^2+...+x_d^2)+igW(x_1,...,x_d)$ on $L^2(\R^d)$, where $W$ is
any odd homogeneous polynomial and $g\in\R$. It is proved that $\P H$ is
self-adjoint and that its eigenvalues coincide (up to a sign) with the singular
values of $H$, i.e. the eigenvalues of $\sqrt{H^\ast H}$. Moreover we
explicitly construct the canonical expansion of $H$ and determine the singular
values $\mu_j$ of $H$ through the Borel summability of their divergent
perturbation theory. The singular values yield estimates of the location of the
eigenvalues \l_j of $H$ by Weyl's inequalities.Comment: 20 page

### Some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant

Comparison between the exact value of the spectral zeta function,
$Z_{H}(1)=5^{-6/5}[3-2\cos(\pi/5)]\Gamma^2(1/5)/\Gamma(3/5)$, and the results
of numeric and WKB calculations supports the conjecture by Bessis that all the
eigenvalues of this PT-invariant hamiltonian are real. For one-dimensional
Schr\"odinger operators with complex potentials having a monotonic imaginary
part, the eigenfunctions (and the imaginary parts of their logarithmic
derivatives) have no real zeros.Comment: 6 pages, submitted to J. Phys.

### Scalar Quantum Field Theory with Cubic Interaction

In this paper it is shown that an i phi^3 field theory is a physically
acceptable field theory model (the spectrum is positive and the theory is
unitary). The demonstration rests on the perturbative construction of a linear
operator C, which is needed to define the Hilbert space inner product. The C
operator is a new, time-independent observable in PT-symmetric quantum field
theory.Comment: Corrected expressions in equations (20) and (21

### $PT$ symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum

Consider in $L^2(R^d)$, $d\geq 1$, the operator family $H(g):=H_0+igW$. \ds
H_0= a^\ast_1a_1+... +a^\ast_da_d+d/2 is the quantum harmonic oscillator with
rational frequencies, $W$ a $P$ symmetric bounded potential, and $g$ a real
coupling constant. We show that if $|g|<\rho$, $\rho$ being an explicitly
determined constant, the spectrum of $H(g)$ is real and discrete. Moreover we
show that the operator \ds H(g)=a^\ast_1 a_1+a^\ast_2a_2+ig a^\ast_2a_1 has
real discrete spectrum but is not diagonalizable.Comment: 20 page

### On the eigenproblems of PT-symmetric oscillators

We consider the non-Hermitian Hamiltonian H=
-\frac{d^2}{dx^2}+P(x^2)-(ix)^{2n+1} on the real line, where P(x) is a
polynomial of degree at most n \geq 1 with all nonnegative real coefficients
(possibly P\equiv 0). It is proved that the eigenvalues \lambda must be in the
sector | arg \lambda | \leq \frac{\pi}{2n+3}. Also for the case
H=-\frac{d^2}{dx^2}-(ix)^3, we establish a zero-free region of the
eigenfunction u and its derivative u^\prime and we find some other interesting
properties of eigenfunctions.Comment: 21pages, 9 figure

### CPT-conserving Hamiltonians and their nonlinear supersymmetrization using differential charge-operators C

A brief overview is given of recent developments and fresh ideas at the
intersection of PT and/or CPT-symmetric quantum mechanics with supersymmetric
quantum mechanics (SUSY QM). We study the consequences of the assumption that
the "charge" operator C is represented in a differential-operator form. Besides
the freedom allowed by the Hermiticity constraint for the operator CP,
encouraging results are obtained in the second-order case. The integrability of
intertwining relations proves to match the closure of nonlinear SUSY algebra.
In an illustration, our CPT-symmetric SUSY QM leads to non-Hermitian polynomial
oscillators with real spectrum which turn out to be PT-asymmetric.Comment: 25 page

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